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Roots of Functions
Background and Commentary
Newton-Raphson and Secant Methods for Finding Roots of Functions
General Background and Review
It is frequently required to find the value of a component of a mathematical equation that satisfies the
equation. For example, for the equation
we may want to find the values of x that ‘satisfy’ the equation. In other words, we want to find those
values of the variable x such that the left side of the equation is equal to 2.
In this topic, we focus on three methods for solving the general equation f(x) = 0. Note that by focusing
on methods for solving the equation f(x) = 0, we do not lose any generality in solving equations because
any equation can be put in the form of f(x) = 0 simply by moving terms from one side of the equality to the
other. For example, while the equation above may not appear to be one which is of the form f(x) = 0, we
can make it such simply by subtracting 2 from both sides of the equation. Thus, the equation above
becomes
We conclude, therefore, that if we can solve an equation of the form f(x) = 0, we can solve any equation.
Note further that values of the unknown x which satisfy the equation f(x) = 0 are referred to as the roots or
zeroes of the function f(x). So that in solving the equation f(x) = 0 for the values of x which satisfy the
equation, we are finding the roots of the function.
Note also that, from a graphical point of view, the roots of the equation f(x) = 0 represent the locations on
a graph of the function where the graph crosses the x axis.
So then, in this topic we will be examining methods for finding the roots of functions of the form f(x) = 0
which translates graphically into methods for finding where the function f(x) = 0 crosses the x axis.
Note that some functions may cross the x-axis more than once, meaning that there are multiple real roots,
as shown on the following graph.
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In addition, some functions may not cross the x-axis at all, as shown on the following graph, meaning that
the function has no real roots. Such functions may have complex or imaginary roots, which we will not
consider at this time.
The three methods (there are other methods in addition to these three) that we will consider are the
Bisection Method (sometimes called the Half-Interval Method), Newton’s Method (also known as the
Newton-Raphson Method), and the Secant Method.
Bisection/Half-Interval Method
The Bisection/Half-Interval method is one of several methods referred to as ‘bracketing methods’ for
finding the roots of a given function. The basic idea in all bracketing methods is to find an interval along
the x-axis (independent variable axis) which contains the root of the function with the interval small
enough that the value of the root can be established to within some desired level of accuracy.
Description
Consider the function represented by the graph below with corresponding values shown in the table.
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x
0
5
10
f(x)
-4.00
2.25
21.00
The bisection method begins by finding an interval [a, b] along the x-axis such that f(a) and f(b) are of
opposite signs; that is, where the product f(a) f(b) < 0. If f(a) and f(b) are of opposite signs, there must be
at least one place along the x-axis where the function crosses the axis; that is, where f(x) = 0.
In the example above, if a = 0 and b = 10, then f(a) = -4.00 and f(b) = 21.00. Since f(a) f(b) = (-4.0)(21.0)
= -84.0 < 0, there must be a root somewhere in the in [a, b] interval.
The next step is to cut the interval in half; that is, let c = ( 0 + 10 ) / 2 = 5.0 for which f(c) = f(5) = 2.25.
Now examine the two new intervals. For the interval [5, 10], f(5) and f(10) are both of the same sign; that
is, f(5) f(10) = (2.25)(21.0) = 47.25 > 0. Therefore, the function does not cross the x-axis between x = 5
and x = 10, so the root must lie in the interval between x = 0 and x = 5.
We can confirm this by calculating the product f(0) f(5) = (-4.0)(2.25) = -9.0 < 0. So that the function does
in fact cross the x-axis somewhere in the interval [0, 5].
If we now re-define the [a, b] interval for a = 0 and b = 5, we can repeat the process until the interval [a, b]
is smaller than some value we choose, such as 10-6. We could then take either the value of a or the
value of b as the value of the root of the function. Alternatively, we could take the average of the final [a,
b] interval as the value for the root; that is, let the root value for x equal (a + b) / 2.
As yet another approach to terminating the iterative procedure, we could choose to terminate when the
value of the function itself at either x = a or x = b is less than some selected value. Since we are finding
the root of the function (that is, the value of x which makes the value of the function equal to zero), we
could choose to terminate when the value of the function is sufficiently close to zero.
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Error Analysis
Let
and
define the starting interval for the bisection method iterations, and let
define the interval size after the i-th interation. Then
and
Thus we can say that the error in the value of the root of the function after n iterations is given by
From this relationship we can also determine the number of iterations needed to satisfy some given error
criterion:
Advantages and Disadvantages
Two advantages of the bisection method (or of any of the bracketing methods in general) are:
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1. the procedure always converges; that is, a root will always be found, provided that at least one root
exists.
2. there is good information available about the error associated with the result.
Two disadvantage of the bisection method are:
1. the method converges relatively slowly.
2. many iterations may be required.
Newton’s (Newton-Raphson) Method
The Newton-Raphson method is one of several methods for finding roots of functions classified as slope
methods. Slope methods use the slope of a straight line to project to the location of a root on the x-axis.
Description
Recall from Calculus that the first derivative of a function represents the slope of the function at any given
point on the curve, as illustrated in the following diagram.
If m represents the slope of the function f(x), then we can write the following relationship:
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Noting that
,
Solving for x1,
Then, similarly,
In general,
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Thus, we have an iterative algorithm such that, given some starting value for x = x0, we can solve the
algorithm repetitively until two successive values of x are within some arbitrary limit; that is, until
Advantages and Disadvantages
The Newton-Raphson method generally provides rapid convergence to the root, provided the initial value
x0 is sufficiently close to the root.
How close is ‘sufficiently close’? That depends on the characteristics of the function itself. As illustrated
by the following graphs, certain initial values of x may cause the solution to diverge or not produce a valid
result.
In example (a) above, a runaway situation is encountered where the solution algorithm does not yield a
finite value for the result.
In example (b) above, a flat spot on the curve is encountered which does not produce an new value of x
which lies on the x-axis. Since the slope of a horizontal line is zero, this situation could also result in a
divide-by-zero computational error, causing a computer program to crash.
In example (c) above, a cyclical or oscillatory situation is encountered where the solution algorithm
produces values of x which cycle back and forth between the same values and never do converge.
Overall, the Newton-Raphson method is a widely used technique which produces good results provided
its limitations are recognized.
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Secant Method
The Secant method is another of the methods for finding roots of functions classified as slope methods.
This method is similar to the Newton-Raphson method but avoids the calculation of derivatives.
Description
While the Newton-Raphson method can be started with only one x-f(x) combination, the secant method
begins with two separate values. Consider the following diagram.
Let m represent the slope of the straight line between two points on the curve of f(x), that is, a secant line
over an arc of the function. Then we can write the following relationship:
Noting that
,
Solving for xn + 1,
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Thus, we have an iterative algorithm such that, given two starting values for x = x0 and x = x1 we can
solve the algorithm repetitively until two successive values of x are within some arbitrary limit; that is, until
Advantages and Disadvantages
The Secant method generally provides fairly rapid convergence to the root but not as rapidly as the
Newton-Raphson method. However, one might note that, except for the starting iteration, the secant
method requires the evaluation of only one function per iteration while the Newton-Raphson method
requires that two functions (the function and its derivative) for each iteration, since the (n + 1) values from
one iteration can be used as the (n) values for the next iteration.
Similar to the Newton-Raphson method, the secant method may also encounter runaway, flat spot, and
cyclical non-convergence characteristics as described above.
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